# Fibonacci sequence

The Fibonacci sequence is the sequence that arises in answer to this problem posed in Fibonacci's
great work *Liber abaci*:

A certain man put a pair of rabbits in a place surrounded on all sides by a wall. How many pairs of rabbits can be produced from that pair in a year if it is supposed that every month each pair begins a new pair which from the second month on becomes productive?

The number of pairs of rabbits in the *n*-th month begins 1, 1, 2,
3, 5, 8, 13, 21, 34, 55, 89, ..., where each term is the sum of the two
terms preceding it. This sequence can be defined recursively as follows: *F*(1) = *F*(2) = 1, *F*(*n* + 1) = *F*(*n*)
+ *F*(*n* - 1) for *n* > 2, where F(*n*) is the *n*-th
Fibonacci number. Johannes Kepler was the
first to point out that the growth rate of the Fibonacci numbers, that is, *F*(*n* + 1) / F(*n*), converges to the golden
ratio, *φ*, which has the value √5 + 1)/2.

In the nineteenth century Fibonacci numbers were discovered in many natural forms.
For example, many types of flower have a
Fibonacci number of petals: daisies tend to
have 34 or 55 petals, while sunflowers have 89 or, in some cases, 144. The seeds of sunflowers spiral outward both to
the left and the right in a Fibonacci number of spirals. Similarly, the
whorls on a pine cone, the numbers of rings on the trunks of palm trees,
the patterns of snails shells, and the genealogy
of the male bee all follow a sequence of Fibonacci numbers. The arrangement
of plant leaves, or phyllotaxis, unfolds to the same pattern because this
results in an optimal solution in terms of the spacing of the leaves or
the amount of light that can reach them. A familiar spiral form, known as
the logarithmic spiral, emerges
when seeds on a plant grow and space themselves according the Fibonacci
sequence. The logarithmic spiral is approximated by the rule: start at the
origin of the Cartesian coordinate system, move *F*(1) units to the
right, move *F*(2) units up, move *F*(3) units to the left, move *F*(4) units down, move *F*(5) units to the right, and so on.
By growing in this way, on structures as such sunflowers, pine cones, and
pineapples, seeds are able to pack themselves together most efficiently.

Fibonacci numbers have so many interesting mathematical properties that
an entire journal, *The Fibonacci Quarterly*, is devoted to them. The
sequence of final digits in Fibonacci numbers repeats in cycles of 60. The
last two digits repeat in 300, the last three in 1500, the last four in
15,000, etc. The product of any four consecutive Fibonacci numbers is the
area of a Pythagorean triangle. The shallow (least steep) diagonals of Pascal's
triangle sum to Fibonacci numbers. Let *m* and *n* be positive
integers, then

*F*(*n*) divides *F*(*mn*)

gcd(*F*(*n*), *F*(*m*)) = *F*(gcd(*m*, *n*)), where "gcd" stands for "greatest common divisor"

(*F*(*n*))^{2} - *F*(*n*+1)*F*(*n*-1)
= (-1)^{n-1}

*F*(1) + *F*(3)
+ *F*(5) + ... + *F*(2*n*-1) = *F*(2*n*)

for every *n*, there are *n* consecutive composite Fibonacci numbers.

An interesting use of the Fibonacci sequence is for converting miles to kilometers. For instance, if you want to know about how many kilometers 5 miles is, take the Fibonacci number (5) and look at the next one (8). 5 miles is about 8 kilometers. This works because it happens that the conversion factor between miles and kilometers is roughly equal to the golden ratio.

The first few Fibonacci numbers that are also prime
numbers are 3, 5, 13, 89, 233, 1597, 28657, 514229, ... It seems likely
that there are infinitely many Fibonacci primes, but this has yet to be
proven. However, it is relatively easy to show that for *n* > or =
4, *u*_{n}+1 is never prime. The Fibonacci sequence
is a special case of the Lucas sequence.

The "tribonacci" series is made by adding the last two digits: 1, 1, 2,
4, 7, 13, 24, 44, 81, ... And from this the quadbonacci series, the pentbonacci
series, and the hexbonacci series, all the way up to the *n*-bonacci
series. Each ratio of successive terms forms a special constant, analogous
to phi.